Harmonic mappings and conformal minimal immersions of Riemann surfaces into $\mathbb{R}^n$

TL;DR

This paper constructs harmonic and minimal immersions of Riemann surfaces into Euclidean spaces, demonstrating the existence of complete, non-decomposable minimal surfaces with prescribed properties and exploring their geometric and conformal structures.

Contribution

It proves the existence of harmonic maps with prescribed Hopf differentials and constructs new classes of complete minimal surfaces with specific geometric features.

Findings

Existence of harmonic maps with given Hopf differential.

Construction of complete minimal surfaces with arbitrary conformal structures.

Examples of non-proper embedded minimal surfaces in higher dimensions.

Abstract

We prove that for any open Riemann surface $N,$ natural number $n\geq 3,$ non-constant harmonic map $h:N\to \mathbb{R}^{n-2}$ and holomorphic 2-form $H$ on $N,$ there exists a weakly complete harmonic map $X=(X_j)_{j=1,\ldots,n}:N \to \mathbb{R}^n$ with Hopf differential $H$ and $(X_j)_{j=3,\ldots,n}=h.$ In particular, there exists a complete conformal minimal immersion $Y=(Y_j)_{j=1,\ldots,n}:N \to \mathbb{R}^n$ such that $(Y_j)_{j=3,\ldots,n}=h.$ As a consequence of these results, complete full non-decomposable minimal surfaces with arbitrary conformal structure and whose generalized Gauss map is non-degenerate and fails to intersect $n$ hyperplanes of $\mathbb{CP}^{n-1}$ in general position are constructed. Moreover, complete non-proper embedded minimal surfaces in $\mathbb{R}^n,$ $\forall n>3,$ are exhibited.